Uniqueness of the Non-Equilibrium Steady State for a 1d BGK Model in Kinetic Theory

Abstract

We continue our investigation of kinetic models of a one-dimensional gas in contact with homogeneous thermal reservoirs at different temperatures. Nonlinear collisional interactions between particles are modeled by a so-called BGK dynamics which conserves local energy and particle density. Weighting the nonlinear BGK term with a parameter $\alpha\in[0,1]$ , and the linear interaction with the reservoirs by $(1-\alpha)$ , we prove that for some $\alpha$ close enough to zero, the explicit spatially uniform non-equilibrium steady state (NESS) is unique, and there are no spatially non-uniform NESS with a spatial density $\rho$ belonging to $L^{p}$ for any $p>1$ . We also show that for all $\alpha\in[0,1]$ , the spatially uniform NESS is dynamically stable, with small perturbation converging to zero exponentially fast.